Mechanical engineering usually deals with forces and their effects on materials. Forces are vectors and so, to study their effects you need to use vector calculus.
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These are the general math courses in an undergraduate program of Mechanical Engineering. Actually, these are also the math courses required in ANY undergraduate Engineering curriculum: Algebra Trigonometry Analytic Geometry Differential Calculus Integral Calculus Mutivariable Calculus Differential Equations
I am an engineering student at the University of Rhode Island. Most schools that offer engineering as a major will require you to go up to Level III Calculus, as well as classes involving equations with multiple unknowns. Mechanical engineering is a very rigorous and difficult major, but if you study hard, you will do absolutely fine.
One uses calculus including differential equations and vector calculus in the undergrad courses which is as far as got.
Vector Algebra and Vector Calculus are used widely in science, especially Physics and engineering.The physical world involves four dimensions, one scalar dimension and three vector dimensions. From this you can say that 3/4 of the world involve vectors.
It is certainly used in calculus, just as calculus can be used in trigonometry.